Cocycle category

In category theory, a branch of mathematics, the cocyle category of objects X, Y in a model category is a category in which the objects are pairs of maps X ← f Z → g Y and the morphisms are obvious commutative diagrams between them. It is denoted by H ( X , Y ) . (It may also be defined using the language of 2-category.)

One has: if the model category is right proper and is such that weak equivalences are closed under finite products,

is bijective.